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On a \(q\)-deformation of modular forms. (English) Zbl 1445.11014

The purpose of this article is to give examples of \(q\)-deformations of hypergeometric evaluations, which are linked with the coefficients of modular forms rather than Dirichlet characters.
Furthermore, several almost \(q\)-hypergeometric congruences modulo (powers of) cyclotomic polynomials in \(q\) are proved or conjectured. It starts with the slowly-converging Ramanujan-type identity
\[\sum_{k=0}^{\infty}\frac{(\frac 12)_k^3}{k!^3}=\frac{\pi}{\Gamma(\frac 34)^4}.\tag{*} \]
A \(q\)-extension of (*) which accommodates the related congruences \(\bmod~p^2\) is given.

MSC:

11B65 Binomial coefficients; factorials; \(q\)-identities
33C05 Classical hypergeometric functions, \({}_2F_1\)
11F03 Modular and automorphic functions
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References:

[1] Aigner, M.; Ziegler, G. M., Proofs from the Book (2018), Springer: Springer Berlin · Zbl 1392.00001
[2] Andrews, G. E., On \(q\)-analogues of the Watson and Whipple summations, SIAM J. Math. Anal., 7, 332-336 (1976) · Zbl 0339.33007
[3] Andrews, G. E.; Berndt, B. C., Ramanujan’s Lost Notebook, Part II (2009), Springer: Springer New York · Zbl 1180.11001
[4] Andrews, G. E.; Berndt, B. C., Ramanujan’s Lost Notebook, Part III (2012), Springer: Springer New York · Zbl 1248.11003
[5] Bailey, W. N., On the analogue of Dixon’s theorem for bilateral basic hypergeometric series, Quart. J. Math. (Oxford) Ser. (2), 1, 318-320 (1950) · Zbl 0038.22801
[6] Baruah, N. D.; Berndt, B. C.; Chan, H. H., Ramanujan’s series for \(1 / \pi \): a survey, Amer. Math. Monthly, 116, 7, 567-587 (2009) · Zbl 1229.11162
[7] Bauer, G., Von den Coefficienten der Reihen von Kugelfunctionen einer Variablen, J. Reine Angew. Math., 56, 101-121 (1859) · ERAM 056.1478cj
[8] Clarke, F. W.; Everitt, W. N.; Littlejohn, L. L.; Vorster, S. J.R., H.J.S. Smith and the Fermat two squares theorem, Amer. Math. Monthly, 106, 7, 652-665 (1999) · Zbl 1029.01006
[9] Cooper, S., Ramanujan’s Theta Functions (2017), Springer: Springer Cham · Zbl 1428.11001
[10] Ekhad, S. B.; Zeilberger, D., A WZ proof of Ramanujan’s formula for \(π\), (Rassias, J. M., Geometry, Analysis, and Mechanics (1994), World Scientific: World Scientific Singapore), 107-108 · Zbl 0849.33003
[11] Gasper, G.; Rahman, M., Basic Hypergeometric Series, Encyclopedia Math. Appl., vol. 96 (2004), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 1129.33005
[12] Guo, V. J.W., A \(q\)-analogue of the (I.2) supercongruence of Van Hamme, Int. J. Number Theory, 15, 1, 29-36 (2019) · Zbl 1461.11037
[13] Guo, V. J.W., Proof of a \(q\)-congruence conjectured by Tauraso, Int. J. Number Theory, 15, 1, 37-41 (2019) · Zbl 1467.11027
[14] Guo, V. J.W., A \(q\)-analogue of a curious supercongruence of Guillera and Zudilin, J. Difference Equ. Appl., 25, 3, 342-350 (2019) · Zbl 1411.11021
[15] Guo, V. J.W., Some \(q\)-congruences with parameters, Acta Arith. (2019), in press, preprint · Zbl 1459.11052
[16] Guo, V. J.W., \(q\)-Analogues of three Ramanujan-type formulas for \(1 / \pi \), Ramanujan J. (2019), in press
[17] Guo, V. J.W.; Schlosser, M. J., Some new \(q\)-congruences for truncated basic hypergeometric series, Symmetry, 11, 2 (2019), Art. 268 · Zbl 1416.11010
[18] Guo, V. J.W.; Schlosser, M. J., Some \(q\)-supercongruences from transformation formulas for basic hypergeometric series (December 2018), 39 pp
[19] Guo, V. J.W.; Zeng, J., Some \(q\)-supercongruences for truncated basic hypergeometric series, Acta Arith., 171, 4, 309-326 (2015) · Zbl 1338.11024
[20] Guo, V. J.W.; Zudilin, W., Ramanujan-type formulae for \(1 / \pi \): \(q\)-analogues, Integral Transforms Spec. Funct., 29, 7, 505-513 (2018) · Zbl 1436.11024
[21] Guo, V. J.W.; Zudilin, W., A \(q\)-microscope for supercongruences, Adv. Math., 346, 329-358 (2019) · Zbl 1464.11028
[22] Jackson, F. H., Certain \(q\)-identities, Quart. J. Math. (Oxford), 12, 167-172 (1941) · Zbl 0063.03007
[23] Levrie, P., Using Fourier-Legendre expansions to derive series for \(1 / \pi\) and \(1 / \pi^2\), Ramanujan J., 22, 2, 221-230 (2010) · Zbl 1203.30005
[24] Mortenson, E., A \(p\)-adic supercongruence conjecture of Van Hamme, Proc. Amer. Math. Soc., 136, 12, 4321-4328 (2008) · Zbl 1171.11061
[25] Ono, K., Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc., 350, 3, 1205-1223 (1998) · Zbl 0910.11054
[26] Ramanujan, S., Modular equations and approximations to \(π\), Quart. J. Math. (Oxford) Ser. (2), 45, 350-372 (1914) · JFM 45.1249.01
[27] Ramanujan, S., Highly composite numbers, Proc. Lond. Math. Soc. (2), 14, 347-409 (1915) · JFM 45.1248.01
[28] Rogers, M.; Wan, J. G.; Zucker, I. J., Moments of elliptic integrals and critical \(L\)-values, Ramanujan J., 37, 1, 113-130 (2015) · Zbl 1383.11048
[29] Schlosser, M. J., \(q\)-Analogues of two product formulas of hypergeometric functions by Bailey, (Nashed, Z.; Li, X., Frontiers in Orthogonal Polynomials and \(q\)-Series (2018), World Scientific: World Scientific Singapore), 445-449 · Zbl 1411.33015
[30] Sun, Z.-H., Congruences concerning Legendre polynomials II, J. Number Theory, 133, 1950-1976 (2013) · Zbl 1277.11002
[31] Van Hamme, L., Some conjectures concerning partial sums of generalized hypergeometric series, \((p\)-Adic Functional Analysis. \(p\)-Adic Functional Analysis, Nijmegen, \(1996. p\)-Adic Functional Analysis. \(p\)-Adic Functional Analysis, Nijmegen, 1996, Lect. Notes Pure Appl. Math., vol. 192 (1997), Dekker: Dekker New York), 223-236 · Zbl 0895.11051
[32] Zudilin, W., Ramanujan-type formulae for \(1 / \pi \): a second wind?, (Yui, N.; etal., Modular Forms and String Duality. Modular Forms and String Duality, Banff, June 3-8, 2006. Modular Forms and String Duality. Modular Forms and String Duality, Banff, June 3-8, 2006, Fields Inst. Commun. Ser., vol. 54 (2008), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 179-188 · Zbl 1159.11053
[33] Zudilin, W., Ramanujan-type supercongruences, J. Number Theory, 129, 1848-1857 (2009) · Zbl 1231.11147
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